Propagation of singularities of Moreau envelopes and distance functions in a Hilbert space

Abstract

The Moreau envelopes of nonconvex functions on a real Hilbert space have points of nondifferentiability called singularities. For the first time in an infinite-dimensional space, the propagation of singularities along intrinsic characteristic curves x(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{x}}(t)$$\\end{document} is studied. The properties of the intrinsic characteristics are investigated in some detail. It is proved that singularities propagate along intrinsic characteristics not only locally but globally in t. Actually, this paper distinguishes between “singularities” and “strong singularities,” two concepts that agree for distance functions but disagree for Moreau envelopes, in general. Along intrinsic characteristics, propagating singularities instantaneously transform into strong singularities. Concerning the distance function to a closed nonempty set E, it is demonstrated that the singular set and ∁E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\complement E$$\\end{document} are homotopy equivalent provided E satisfies a certain condition which is weaker than the boundedness of ∁E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\complement E$$\\end{document}. The notion of intrinsic characteristics was introduced by Cannarsa and Cheng (Calc Var Part Differ Equ 56, 2017).